Sultan's Dowry

A sticky problem in probability that
first came to light in Martin Gardner's
"Mathematical Recreations" column in the February 1960 issue of Scientific
American. Gardner's original version has become known as the Secretary
Problem. In the exactly equivalent form called the Sultan's Dowry Problem,
a sultan has granted a commoner the chance to marry one of his hundred daughters.
The commoner will be shown the daughters one at a time and will be told
each daughter's dowry. The commoner has only one chance to accept or reject
each daughter; he can't go back and choose one that he has previously rejected.
The sultan's catch is that the commoner may only marry the daughter with
the highest dowry. What is the commoner's best strategy, assuming that he
knows nothing in advance about the way the dowries are distributed?

Many mathematicians have tackled this question and numerous papers have
been written on the subject. It has even spawned its own area of study within
the field of management science. The consensus among those who have worked
on the problem is that the commoner's best strategy is to let a certain
fraction of the daughters pass and then choose the next one who has a dowry
higher than any of the ones seen up to that point. The exact number to skip
is determined by the condition that the odds that the highest dowry has
already been seen is just greater than the odds that it remains to be seen
and that if it is seen it will be picked. This amounts to finding
the smallest x such that:

x/n > x/n × [1/(x+1)
+ ... + 1/(n-1)].

Substituting n = 100 leads to the conclusion that the commoner
should wait until he has seen 37 of the daughters, then pick the first daughter
with a dowry that is bigger than any that have already been revealed. With
this strategy, his odds of choosing the daughter with the highest dowry
are surprisingly high: about 37%.